1. Parity and Other Option Relationships
Chapter 9
Parity and Other Option Relationships

2. IBM Option Quotes
Table 9.1 IBM option prices, dollars per share, October 16, The closing price of IBM on that day was $
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3. Call – put = PV (forward price – strike price)
Put-Call Parity
For European options with the same strike price and time to expiration the parity relationship is
Call – put = PV (forward price – strike price) or
Intuition
Buying a call and selling a put with the strike equal to the forward price (F0,T = K) creates a synthetic forward contract and hence must have a zero price
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4. Parity for Options on Stocks
If underlying asset is a stock and Div is the dividend stream, then e-rT F0,T = S0 – PV0,T (Div), therefore
Rewriting above
For index options, , therefore
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5. Parity for Options on Stocks (cont’d)
Examples 9.1 & 9.2
Price of a non-dividend paying stock: $40, r=8%, option strike price: $40, time to expiration: 3 months, European call: $2.78, European put: $ $2.78=$1.99+$40 – $40e -0.08x0.25
Additionally, if the stock pays $5 just before expiration, call: $0.74, and put: $ $0.74=$4.85+($40 – $5e-0.08x0.25) – $40e-0.08x0.25
Synthetic security creation using parity
Synthetic stock: buy call, sell put, lend PV of strike and dividends
Synthetic T-bill: buy stock, sell call, buy put (conversion)
Synthetic call: buy stock, buy put, borrow PV of strike and dividends
Synthetic put: sell stock, buy call, lend PV of strike and dividends
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6. Generalized Parity Relationship
Table Payoff table demonstrating that there is an arbitrage opportunity unless −C(St , Qt , T − t) + P(St , Qt, T − t) + F P t ,T (S) − FP t ,T (Q) = 0.
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7. Properties of Option Prices
American versus European
Since an American option can be exercised at anytime, whereas a European option can only be exercised at expiration, an American option must always be at least as valuable as an otherwise identical European option
CAmer(S, K, T) > CEur(S, K, T)
PAmer(S, K, T) > PEur(S, K, T)
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8. Properties of Option Prices (cont’d)
Option price boundaries
Call price cannot
be negative
exceed stock price
be less than price implied by put-call parity using zero for put price:
Put price cannot
be more than the strike price
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9. Properties of Option Prices (cont’d)
Early exercise of American options
A non-dividend paying American call option should not be exercised early, because
That means, one would lose money be exercising early instead of selling the option
If there are dividends, it may be optimal to exercise early
It may be optimal to exercise a non-dividend paying put option early if the underlying stock price is sufficiently low
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10. Properties of Option Prices (cont’d)
Time to expiration
An American option (both put and call) with more time to expiration is at least as valuable as an American option with less time to expiration. This is because the longer option can easily be converted into the shorter option by exercising it early
A European call option on a non-dividend paying stock will be more valuable than an otherwise identical option with less time to expiration.
European call options on dividend-paying stock and European puts may be less valuable than an otherwise identical option with less time to expiration
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11. Properties of Option Prices (cont’d)
Different strike prices (K1 < K2 < K3), for both European and American options
A call with a low strike price is at least as valuable as an otherwise identical call with higher strike price
A put with a high strike price is at least as valuable as an otherwise identical call with low strike price
The premium difference between otherwise identical calls with different strike prices cannot be greater than the difference in strike prices
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12. Properties of Option Prices (cont’d)
Different strike prices (K1 < K2 < K3), for both European and American options
The premium difference between otherwise identical puts with different strike prices cannot be greater than the difference in strike prices
Premiums decline at a decreasing rate for calls with progressively higher strike prices. (Convexity of option price with respect to strike price)
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13. Summary of Parity Relationships
Table 9.4 Versions of put-call parity. Notation in the table includes the spot currency exchange rate, x0; the risk-free interest rate in the foreign currency, rf ; and the current bond price, B0.
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14. Chapter 9
Additional Art

15. Equation 9.1
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16. Equation 9.2
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17. Figure 9.1 Cash flows for outright purchase of stock and for synthetic stock created by buying a 40-strike call and selling a 40-strike put.
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18. Equation 9.3
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19. Equation 9.4
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20. Equation 9.5
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21. Equation 9.6
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22. Table 9.3 The equivalence of buying a dollar-denominated euro call and a euro-denominated dollar put. In transaction I, we buy dollar-denominated call options permitting us to buy =C1 for a strike price of $0.92. In transaction II, we buy one euro denominated put permitting us to sell $1 for a strike price of =C = =C The option premium is =C
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23. Equation 9.7
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24. Equation 9.8a and Equation 9.8b
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25. Equation 9.9
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26. Equation 9.10
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27. Equation 9.11
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28. Equation 9.12
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29. Equation 9.13
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30. Equation 9.14
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31. Equation 9.15
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32. Equation 9.16
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33. Equation 9.17
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34. Equation 9.18
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35. Figure 9.2 Profit from purchasing a 50-strike call for $9 and selling a 55-strike call for $10. This is an arbitrage.
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36. Figure 9.3 Profit from selling a 50- strike call for $18 and buying a 55-strike call for $12. This is an arbitrage.
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37. Figure 9.4 Profit from purchasing a 50-strike put for $9 and selling a 55-strike put for $15. This is an arbitrage.
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38. Figure 9.5 Profit from purchasing a 50-strike put for $4, selling two 55-strike puts for $8, and buying a 60-strike put for $11. This is an arbitrage.
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